Data communication systems comprise three basic components: a transmitter; a transmission channel; and a receiver. Transmitted data become altered due to noise corruption and channel distortion. To reduce the presence of errors caused by noise corruption and channel distortion, redundancy is intentionally introduced, and the receiver uses a decoder to make corrections. In modern data communication systems, the use of error correction codes plays a fundamental role in achieving transmission accuracy, as well as in increasing spectrum efficiency. Using error correction codes, the transmitter encodes the data by adding parity check information and sends the encoded data through the transmission channel to the receiver. The receiver uses the decoder to decode the received data and to make corrections using the added parity check information.
Binary Low Density Parity Check (LDPC) codes were first disclosed in R. G. Gallager: “Low Density Parity Check Codes”, Cambridge, Mass.: MIT Press, 1963. However, due to their encoding and decoding complexity, these codes were ignored for over 30 years until MacKay revived interest in these codes in the late 90's. In D. MacKay: “Good error-correcting codes based on very sparse matrices”, IEEE Trans. Inf. Theory, Vol. 45, No. 2, pp. 399-431, 1999, MacKay showed that LDPC codes have good performance approaching the Shannon limit for large block lengths.
In addition to using longer codes, performance of LDPC codes is also improved by using non-binary LDPC codes, capable of increasing the error correcting capability. Binary LDPC codes have parity check matrices with elements from a Galois Field GF(2) and use modulo-2 arithmetic, while non-binary LDPC codes are defined over GF(q). For example, reference M. Davey and D. MacKay: “Low-density parity check codes over GF(q)”, IEEE Commun. Lett., Vol. 2, No. 6, pp. 165-167, 1998, has shown that non-binary LDPC codes have superior performance to binary codes with identical block length computed in bits.
In reference S. Sharifi Tchrani, W. Gross, and S. Mannor: “Stochastic decoding of LDPC codes”, IEEE Commun. Lett., Vol. 10, No. 10, pp. 716-718, 2006, a binary LDPC decoder based on stochastic computing is disclosed. A major advantage of using a stochastic decoder is a substantially simplified circuitry.
It would be highly desirable to provide a decoding method and system for stochastic decoding of non-binary LDPC codes.